Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.     

Convergence of a Stochastic Method for the Modeling of Polymeric Fluids

We present a convergence analysis of a stochastic method for numerical modeling of complex fluids using Brownian configuration fields (BCF) for shear flows. The analysis takes into account the special structure of the stochastic partial differential equations for shear flows. We establish the optimal rate of convergence. We also analyze the nature of the error by providing its leading order asymptotics.     

A string model of “exotic” particles

The bosonic string is investigated using the approach to 4D string dynamics previously proposed by the author. The physical states and the mass spectrum are constructed. The scale invariance of the theory results in the linear dependence of the squared mass μ² on the “spin” . Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 223–244 May. 1998.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).     

Method of integral equations in the scalar problem of diffraction on a system consisting of a “soft” and a “hard” screen and an inhomogeneous body

We consider the scalar problem on the diffraction of a plane wave on a system of two screens with boundary conditions of the first and the second kind and a solid inhomogeneous body in the semiclassical setting. The original boundary value problem for the Helmholtz equation is reduced to a system of singular integral equations over the body domain and the screen surfaces. We prove the equivalence of the integral and differential statements of the problem, the solvability of the system of integral equations in Sobolev spaces, and the smoothness of its solutions. To solve the integral equations approximately, we use the Bubnov-Galerkin method; we introduce basis functions on the body and the screens and prove the consistency and convergence of the numerical method.     

Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system

A conic linear system is a system of the form?P(d): find x that solves b - Ax∈C _Y , x∈C _X ,? where C _X and C _Y are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c ₁ C(d), and such that r≥ and R≤c ₃ C(d), where c ₁,c ₂,c ₃ are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small. Received November 2, 1995 / Revised version received June 26, 1998?Published online May 12, 1999     

Modeling,shape analysis and computation of the equilibrium pore shape near a PEM–PEM intersection

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: (i) the existence and uniqueness of the electrical potential, (ii) the shape derivatives of state variables and (iii) the shape differentiability of the corresponding energy and the corresponding Euler–Lagrange equation. The latter leads to a modified Young–Laplace equation on the interface. This modified equation is compared with the classical Young–Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.     

Transferring a system with an unknown disturbance under the optimal control to a state of dynamic balance and to an ε-vicinity of the final state

A problem of transferring a linear system to a state of dynamic balance under a simultaneous action of an unknown disturbance and time-optimal control is considered. The optimal control is calculated along the phase trajectory, and it is periodically updated for discrete phase coordinate values. It is proved that the phase trajectory comes to the dynamic equilibrium point and makes undamped periodic motions (a stable limit cycle). The location of the dynamic equilibrium point and the limit cycle form are considered as functions of various parameters. With the disturbance calculated in the process of control, the accuracy of transferring to the required final state increases. A method for estimating attainable accuracy is presented. Results of simulation and numerical calculations are given.     

The Use of the “Golden Section” Rule in Probability Modeling Algorithms

In this article, we consider an algorithm for solving the quadratic assignment problem. The suggested golden section method is a modification of the accelerated probability modeling A3Y algorithm.     

Singularity of the “swallow-tail” type and the glass-glass transition in a system of collapsing hard spheres

In the mode-coupling approximation, we consider the transition to the glass state in a system of collapsing hard spheres (a system with the hard-core potential to which a repulsive step is added). We propose an approximation for the structure factor of the system, which we use to construct the phase diagram of the transition to the glass state. We show that there exists a maximum on the liquid-glass curve corresponding to the reentrant transition to the glass state in the system. In the framework of the proposed model, we consider bifurcations of solutions of the equations describing the transition to the glass state and show that there exist bifurcations of the “swallow-tail” type corresponding to the glass-glass transition.     

Commons dilemma and the “Tragedy of the Commons”

There are a lot of studies on commons dilemma. But few studies succeed to formulate it adequately. Moreover, although there are some situations where the dilemma does not occur, these studies focuses only the case that the dilemma does occur. In the present paper, we formulate the commons dilemma as an N‐person game and examine what situations our formulation represents. We show that there are six regions with respect to the value decrement induced by one additional animal and that commons dilemma does not occur in some of these regions.     

Erratum to “On a Classical Theorem on the Diameter and Minimum Degree of a Graph”

The original version of the article was published in[1]. Unfortunately, the original version of this article contains a mistake:in Theorem 6.2 appears that β(n, Δ)=(n-Δ+5)/4 but the correct statement is β(n, Δ)=(n-Δ+4)/4. In this erratum we correct the theorem and give the correct proof.     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

Modeling the interaction of an electromagnetic field with a two-dimensional surface in the framework of symanzik’s approach

Symanzik’s approach to the construction of models for the interaction between quantum fields and macroobjects allows finding the general form of the action functional with a Chern-Simons potential up to an arbitrary dimensionless constant; it is used to describe the interaction of a material surface with the electromagnetic field. We discuss results obtained in static models of such a type. We also consider a simple dynamic model of the interaction of a massless scalar field with moving planes.     

Numbers of the Equitype and Quasi-Equitype Decompositions of a Permutation

In the paper“The equitype and quasi-equitype decompositions of arbitrarypermutation by permutations of order 2 or 3”,we proved that any even permu-tation can be expressed as a product of two equitype involutions and any odd per-mutation can be expressed as a product of two quasi-equitype involutions. Meanwh-ile,we gave the condition on t such that an even permutation has(2~t)equitypedecompositions and the condition on t such that an odd permutation has(2;2~t)quasi-equitype decompositions.In this paper,we give the numbers of the equitypeand quasi-equitype decompositions of a permutation.The main results are